A musical tuning which is not just-intonation; that is, the intervals are not small-integer ratios.
The most familiar types of temperament today are the equal temperaments, but historically many temperaments have also been unequal; all well-temperaments, and all those belonging to the meantone family, fall into the latter category.
I would prefer:
A tuning which approximates just-intonation; that is, the intervals are close to small-integer ratios, and the scale is somehow simplified relative to its just equivalent.
Some tunings make no attempt to approximate JI, and so should not be considered temperaments.
A tuning system which contains tempered intervals.
I would say a regular temperament is determined by a homomorphic mapping from the p-limit, or possibly another finitely-generated subgroup of the positive rationals, to an abstract free group of smaller rank. This can be specified (uniquely determined) by giving a wedgie, a kernel, or an explicit mapping.
Note I do not include the tuning map as part of the definition, so this is an abstract definition of what a regular temperament is. However, this is how the word is most commonly used; people may object to it but the same people talk of 1/4-comma meantone or 2/7-comma meantone as if they were both meantone. My definition also says that even though 31-et meantone is tuned to a group of rank one, it is still qua meantone a group of rank two, and the tuning mapping is another issue.
You might also note that this definition, which says a temperament is a morphism, makes no sense unless you get the category right, which connects to the thread about spaces.
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